I've never heard that phrase before but a jemmy is a crowbar so I suppose as force is involved, jemmy seems to have been changed to Jimmy. Mind you it could be that jemmy derives from Jimmy, in which case I have no idea
Just had a look in Chambers Dictionary online and indeed, jemmy derives from the name James but does not explain why. Could it be the guy who made it popular?
Certainly in parts of Scotland a jimmy is a gym-shoe...ie a piece of footwear...so I'd guess your phrase is just a variant of 'Give it some wellie' (wellington-boot), another type of footwear. That means put your foot down or speed up or hit it harder or put more effort into what you're doing etc.